Biased Sampling and Confidence

DMS 9626347 Woodroofe 该研究涉及有偏抽样模型， 指数时间序列 在有偏抽样模型中， 在研究中纳入受试者可能取决于以下变量 兴趣-例如，主题的大小。 兴趣集中在 包含概率取决于以下变量的情况 以单调的方式感兴趣，但不假设包含 概率是已知的或已知几个参数。 的 研究包括开发偏见存在的测试， 在存在偏差时制定估计程序， 研究检验和估计的性质。 可能性 和惩罚可能性被用来开发测试， 估计，并通过组合的性质进行了研究 渐近分析和模拟。 指数时间序列是一个 有限维分布形成的随机过程 指数家族 对于这样的过程， 在相当一般的条件下，归一化极大似然估计量是渐近正态的。 研究涉及 开发这些采样的高阶近似 分布，并使用精细近似来形成 修正的置信度。 从数学上讲，结果的形式是 使用贝叶斯方法的非常弱的扩展 以获得抽样分布的近似值。 在涉及以下问题的调查中，经常会出现偏倚样本： 寻找隐藏的物体，因为找到一个 对象可能取决于其属性。 例如，天文学家 更有可能找到一个大的明亮星系，而不是一个小的昏暗星系。 一个，地质学家更有可能找到一个大油井，而不是一个 小的。 以前关于这类问题的工作主要集中在 包含概率依赖于变量的情况 以已知的方式感兴趣。 这项新研究涉及开发 数据分析是适当的，当后者 关系是未知的，包括方法，估计 与观测数据的关系。 关系很重要， 因为没有找到的物体的数量取决于它在关键的 路上了 该研究考虑指数时间序列，其中包括经典的时间序列，如股票价格和天气，但也 许多其他人喜欢自适应设计的实验（实验 他们自己设计的）和连续的临床试验。

DMS 9626347 Woodroofe The research concerns biased sampling models and exponential time series. In biased sampling models, the probability of including a subject in the study may depend on variables of interest--for example, the size of the subject. Interest centers on cases in which the inclusion probabilities depend on the variables of interest in a monotone way, but without assuming that the inclusion probabilities are known or known up to a few parameters. The research involves developing tests for the presence of bias, developing estimation procedures when bias is present and studying the properties of both tests and estimators. Likelihood and penalized likelihood are used to develop the tests and estimators, and the properties are studied through a combination of asymptotic analysis and simulation. An exponential time series is a stochastic process whose finite dimensional distributions form exponential families. For such process, the sampling distributions of normalized maximum likelihood estimators are asymptotically normal, under quite general conditions. The research involves developing higher order approximations to the these sampling distributions and using the refined approximations to form corrected confidence sets. Mathematically, the results take the form of very weak expansions in which a Bayesian approach is used to obtain approximations to sampling distributions. Biased samples arise frequently in investigations that involve searching for hidden objects, since the probability of finding an object may depend on its properties. For example, astronomers are more likely to find a large bright galaxy than a small dim one, and geologists are more likely to find a large oil well than a small one. Previous work on such problems has concentrated on the case in which the inclusion probabilities depend on the variables of interest in a known way. The new research involves developing data analyses that are appr opriate when the latter relationship is not known and includes methods for estimating the relationship from observed data. The relationship is important, because the number of objects not found depends on it in crucial way. The research considers exponential time series which include classical time series, like stock prices and weather, but also many others like adaptively designed experiments (experiments that design themselves) and sequential clinical trials.